# Properties

 Label 216.2.d Level 216 Weight 2 Character orbit d Rep. character $$\chi_{216}(109,\cdot)$$ Character field $$\Q$$ Dimension 16 Newform subspaces 3 Sturm bound 72 Trace bound 4

# Related objects

## Defining parameters

 Level: $$N$$ = $$216 = 2^{3} \cdot 3^{3}$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 216.d (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ = $$8$$ Character field: $$\Q$$ Newform subspaces: $$3$$ Sturm bound: $$72$$ Trace bound: $$4$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(216, [\chi])$$.

Total New Old
Modular forms 42 16 26
Cusp forms 30 16 14
Eisenstein series 12 0 12

## Trace form

 $$16q + 2q^{4} + O(q^{10})$$ $$16q + 2q^{4} + 2q^{10} + 18q^{16} - 26q^{22} - 16q^{25} - 14q^{28} + 8q^{31} - 20q^{34} - 30q^{40} + 4q^{46} + 24q^{49} - 8q^{52} - 16q^{55} + 28q^{58} - 10q^{64} + 66q^{70} - 8q^{73} + 80q^{76} - 72q^{79} - 52q^{82} + 70q^{88} - 48q^{94} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(216, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
216.2.d.a $$4$$ $$1.725$$ $$\Q(\zeta_{8})$$ $$\Q(\sqrt{-6})$$ $$0$$ $$0$$ $$0$$ $$-4$$ $$q+\zeta_{8}^{2}q^{2}-2q^{4}+(\zeta_{8}-\zeta_{8}^{2})q^{5}+(-1+\cdots)q^{7}+\cdots$$
216.2.d.b $$4$$ $$1.725$$ $$\Q(i, \sqrt{7})$$ None $$0$$ $$0$$ $$0$$ $$4$$ $$q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}-\beta _{3}q^{5}+q^{7}+\cdots$$
216.2.d.c $$8$$ $$1.725$$ 8.0.629407744.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+\beta _{2}q^{4}+\beta _{4}q^{5}+(-\beta _{2}-\beta _{6}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(216, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(216, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(24, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(72, [\chi])$$$$^{\oplus 2}$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ ($$( 1 + 2 T^{2} )^{2}$$)($$1 - 3 T^{2} + 4 T^{4}$$)($$1 - 2 T^{2} + 2 T^{4} - 8 T^{6} + 16 T^{8}$$)
$3$ 1
$5$ ($$1 + 2 T^{2} - 21 T^{4} + 50 T^{6} + 625 T^{8}$$)($$( 1 - 9 T^{2} + 25 T^{4} )^{2}$$)($$( 1 - 8 T^{2} + 38 T^{4} - 200 T^{6} + 625 T^{8} )^{2}$$)
$7$ ($$( 1 + 2 T - 3 T^{2} + 14 T^{3} + 49 T^{4} )^{2}$$)($$( 1 - T + 7 T^{2} )^{4}$$)($$( 1 + 7 T^{2} + 49 T^{4} )^{4}$$)
$11$ ($$1 - 10 T^{2} - 21 T^{4} - 1210 T^{6} + 14641 T^{8}$$)($$( 1 - 13 T^{2} + 121 T^{4} )^{2}$$)($$( 1 - 24 T^{2} + 358 T^{4} - 2904 T^{6} + 14641 T^{8} )^{2}$$)
$13$ ($$( 1 - 13 T^{2} )^{4}$$)($$( 1 + 2 T^{2} + 169 T^{4} )^{2}$$)($$( 1 - 30 T^{2} + 451 T^{4} - 5070 T^{6} + 28561 T^{8} )^{2}$$)
$17$ ($$( 1 + 17 T^{2} )^{4}$$)($$( 1 + 6 T^{2} + 289 T^{4} )^{2}$$)($$( 1 + 16 T^{2} + 614 T^{4} + 4624 T^{6} + 83521 T^{8} )^{2}$$)
$19$ ($$( 1 - 19 T^{2} )^{4}$$)($$( 1 - 10 T^{2} + 361 T^{4} )^{2}$$)($$( 1 - 18 T^{2} + 691 T^{4} - 6498 T^{6} + 130321 T^{8} )^{2}$$)
$23$ ($$( 1 + 23 T^{2} )^{4}$$)($$( 1 + 18 T^{2} + 529 T^{4} )^{2}$$)($$( 1 + 16 T^{2} - 250 T^{4} + 8464 T^{6} + 279841 T^{8} )^{2}$$)
$29$ ($$( 1 - 50 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 22 T^{2} + 841 T^{4} )^{2}$$)($$( 1 - 36 T^{2} + 1558 T^{4} - 30276 T^{6} + 707281 T^{8} )^{2}$$)
$31$ ($$( 1 - 10 T + 69 T^{2} - 310 T^{3} + 961 T^{4} )^{2}$$)($$( 1 + 7 T + 31 T^{2} )^{4}$$)($$( 1 - 2 T + 31 T^{2} )^{8}$$)
$37$ ($$( 1 - 37 T^{2} )^{4}$$)($$( 1 - 46 T^{2} + 1369 T^{4} )^{2}$$)($$( 1 - 102 T^{2} + 4891 T^{4} - 139638 T^{6} + 1874161 T^{8} )^{2}$$)
$41$ ($$( 1 + 41 T^{2} )^{4}$$)($$( 1 + 54 T^{2} + 1681 T^{4} )^{2}$$)($$( 1 + 4 T^{2} + 1574 T^{4} + 6724 T^{6} + 2825761 T^{8} )^{2}$$)
$43$ ($$( 1 - 43 T^{2} )^{4}$$)($$( 1 + 26 T^{2} + 1849 T^{4} )^{2}$$)($$( 1 - 108 T^{2} + 6166 T^{4} - 199692 T^{6} + 3418801 T^{8} )^{2}$$)
$47$ ($$( 1 + 47 T^{2} )^{4}$$)($$( 1 + 47 T^{2} )^{4}$$)($$( 1 + 80 T^{2} + 3750 T^{4} + 176720 T^{6} + 4879681 T^{8} )^{2}$$)
$53$ ($$1 - 94 T^{2} + 6027 T^{4} - 264046 T^{6} + 7890481 T^{8}$$)($$( 1 - 25 T^{2} + 2809 T^{4} )^{2}$$)($$( 1 - 132 T^{2} + 9526 T^{4} - 370788 T^{6} + 7890481 T^{8} )^{2}$$)
$59$ ($$( 1 + 10 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 - 102 T^{2} + 3481 T^{4} )^{2}$$)($$( 1 - 56 T^{2} + 6374 T^{4} - 194936 T^{6} + 12117361 T^{8} )^{2}$$)
$61$ ($$( 1 - 61 T^{2} )^{4}$$)($$( 1 - 61 T^{2} )^{4}$$)($$( 1 - 46 T^{2} - 1101 T^{4} - 171166 T^{6} + 13845841 T^{8} )^{2}$$)
$67$ ($$( 1 - 67 T^{2} )^{4}$$)($$( 1 - 67 T^{2} )^{4}$$)($$( 1 - 125 T^{2} + 4489 T^{4} )^{4}$$)
$71$ ($$( 1 + 71 T^{2} )^{4}$$)($$( 1 - 110 T^{2} + 5041 T^{4} )^{2}$$)($$( 1 + 71 T^{2} )^{8}$$)
$73$ ($$( 1 + 14 T + 123 T^{2} + 1022 T^{3} + 5329 T^{4} )^{2}$$)($$( 1 - 3 T + 73 T^{2} )^{4}$$)($$( 1 - 2 T + 35 T^{2} - 146 T^{3} + 5329 T^{4} )^{4}$$)
$79$ ($$( 1 + 10 T + 79 T^{2} )^{4}$$)($$( 1 - 4 T + 79 T^{2} )^{4}$$)($$( 1 + 12 T + 187 T^{2} + 948 T^{3} + 6241 T^{4} )^{4}$$)
$83$ ($$1 + 134 T^{2} + 11067 T^{4} + 923126 T^{6} + 47458321 T^{8}$$)($$( 1 - 117 T^{2} + 6889 T^{4} )^{2}$$)($$( 1 - 236 T^{2} + 25910 T^{4} - 1625804 T^{6} + 47458321 T^{8} )^{2}$$)
$89$ ($$( 1 + 89 T^{2} )^{4}$$)($$( 1 + 66 T^{2} + 7921 T^{4} )^{2}$$)($$( 1 + 304 T^{2} + 38918 T^{4} + 2407984 T^{6} + 62742241 T^{8} )^{2}$$)
$97$ ($$( 1 + 2 T - 93 T^{2} + 194 T^{3} + 9409 T^{4} )^{2}$$)($$( 1 - 7 T + 97 T^{2} )^{4}$$)($$( 1 + 6 T + 175 T^{2} + 582 T^{3} + 9409 T^{4} )^{4}$$)